CFT Cheatsheet

This appendix collects the conformal field theory facts used repeatedly in the course. It is not a substitute for a full CFT course. It is a working reference for AdS/CFT calculations: which quantities are fixed by symmetry, which quantities are genuine dynamical data, and how those data become bulk fields, masses, couplings, and semiclassical gravity.

A CFT is a quantum field theory invariant under transformations that preserve angles. In flat Euclidean space, the global conformal group is $SO(d+1,1)$ for $d$-dimensional CFTs. In Lorentzian signature, it is $SO(2,d)$. The local group becomes much larger in $d=2$, where the stress tensor generates two Virasoro algebras.

A CFT is specified by its spectrum of primaries and OPE coefficients, subject to symmetry, Ward identities, unitarity, and crossing. In holographic CFTs, the same data reorganize into bulk fields, masses, interactions, and eventually a semiclassical spacetime description.

One-line summary

The intrinsic local data of a CFT are

$$\boxed{ \text{CFT data} = \left\{ \Delta_i,\ell_i,R_i;\ C_{ijk} \right\} \quad \text{subject to crossing, unitarity, and Ward identities.} }$$

Here $\Delta_i$ are scaling dimensions, $\ell_i$ are spin or Lorentz-representation data, $R_i$ are global-symmetry representations, and $C_{ijk}$ are OPE coefficients after choosing operator normalizations.

In a holographic large-$N$ CFT,

$$\text{single-trace primary} \longleftrightarrow \text{single-particle bulk field},$$ $$\Delta \longleftrightarrow \text{bulk mass}, \qquad C_{ijk} \longleftrightarrow \text{bulk interaction strength}, \qquad C_T \longleftrightarrow \frac{L^{d-1}}{G_N}.$$

Coordinates and conventions

Unless stated otherwise, this appendix uses flat Euclidean coordinates $x^i$, $i=1,\ldots,d$, with

$$x_{ij}=x_i-x_j, \qquad x_{ij}^2=(x_i-x_j)^2, \qquad |x_{ij}|=\sqrt{x_{ij}^2}.$$

Lorentzian formulas require operator-ordering and $i\epsilon$ prescriptions. Euclidean, time-ordered, retarded, advanced, and Wightman correlators are different objects. For the real-time dictionary, use the Green Functions Cheatsheet together with the real-time holography pages.

A conformal map changes the metric by a local Weyl factor:

$$\delta_{ij}\,dx'^i dx'^j = \Omega(x)^2\delta_{ij}\,dx^i dx^j.$$

A scalar primary of scaling dimension $\Delta$ transforms as

$$\mathcal O'(x') = \Omega(x)^{-\Delta}\mathcal O(x).$$

For spinning operators, the transformation also includes the local rotation induced by the conformal map.

Conformal Killing vectors

An infinitesimal conformal transformation is generated by a vector field $\xi^i(x)$ satisfying

$$\partial_i\xi_j+ \partial_j\xi_i = \frac{2}{d}\left(\partial\cdot\xi\right)\delta_{ij}.$$

For $d>2$, the independent global transformations are:

transformation	infinitesimal form	generator
translations	$\xi^i=a^i$	$P_i$
rotations	$\xi^i=\omega^i{}_j x^j$	$M_{ij}$
dilatations	$\xi^i=\lambda x^i$	$D$
special conformal transformations	$\xi^i=b^i x^2-2x^i b\cdot x$	$K_i$

A useful algebra convention is

$$[D,P_i]=P_i, \qquad [D,K_i]=-K_i,$$ $$[K_i,P_j]=2\delta_{ij}D-2M_{ij},$$

with the standard rotation commutators for $M_{ij}$. Some authors include factors of $i$ depending on whether generators are chosen Hermitian or anti-Hermitian. The physics is in the representation theory: $P_i$ raises scaling dimension, $K_i$ lowers it, and primaries are annihilated by $K_i$ at the origin.

Primaries, descendants, and conformal multiplets

A local operator $\mathcal O(0)$ is a primary if it is annihilated by special conformal generators at the origin:

$$[K_i,\mathcal O(0)]=0.$$

It is an eigenoperator of dilatations:

$$[D,\mathcal O(0)]=\Delta\mathcal O(0).$$

Descendants are obtained by acting with translations:

$$P_{i_1}\cdots P_{i_n}\mathcal O(0) \quad \longleftrightarrow \quad \partial_{i_1}\cdots\partial_{i_n}\mathcal O(0).$$

The conformal multiplet generated from a primary is the CFT analogue of a highest-weight representation. The data attached to a primary are often summarized as

$$\mathcal O_i: \qquad (\Delta_i,\ell_i,R_i),$$

where $\Delta_i$ is the scaling dimension, $\ell_i$ denotes spin or Lorentz representation data, and $R_i$ denotes global-symmetry representation data.

In AdS/CFT, a primary conformal multiplet corresponds to a bulk one-particle field together with its AdS descendants. The descendants are not new bulk species; they are excitations of the same bulk field.

Radial quantization and the cylinder

On $\mathbb R^d\setminus\{0\}$, write

$$r=e^\tau, \qquad x^i=rn^i, \qquad n^i n_i=1.$$

Then

$$ds_{\mathbb R^d}^2 =dr^2+r^2d\Omega_{d-1}^2 =e^{2\tau}\left(d\tau^2+d\Omega_{d-1}^2\right).$$

Because a CFT is insensitive to Weyl rescaling up to anomaly terms, flat space is conformally equivalent to the cylinder:

$$\mathbb R^d\setminus\{0\} \cong \mathbb R_\tau\times S^{d-1}.$$

The cylinder Hamiltonian is the dilatation operator:

$$H_{\mathrm{cyl}}=D.$$

A primary operator creates a cylinder energy eigenstate:

$$\mathcal O(0)|0\rangle \longleftrightarrow |\mathcal O\rangle, \qquad H_{\mathrm{cyl}}|\mathcal O\rangle = \Delta |\mathcal O\rangle.$$

This is the CFT side of the global AdS statement that a free scalar field with dimension $\Delta$ has normal-mode frequencies

$$\omega_{n,\ell}=\Delta+2n+\ell.$$

Unitarity bounds in $d\ge3$

For unitary CFTs in $d\ge3$, primary dimensions obey lower bounds. For symmetric traceless tensor primaries:

operator type	bound	saturation means
scalar, non-identity	$\Delta\ge \dfrac{d-2}{2}$	free scalar equation when saturated
spin $\ell\ge1$	$\Delta\ge \ell+d-2$	conserved current when saturated
spinor	$\Delta\ge \dfrac{d-1}{2}$	free fermion equation when saturated

For spin $\ell=1$, saturation gives

$$\Delta=d-1, \qquad \partial_\mu J^\mu=0.$$

For spin $\ell=2$, saturation gives

$$\Delta=d, \qquad \partial_\mu T^{\mu\nu}=0.$$

In holography, these shortening conditions correspond to gauge redundancies in the bulk: conserved currents map to gauge fields, and the conserved stress tensor maps to the graviton.

Two-point functions

For scalar primaries,

$$\langle \mathcal O_i(x)\mathcal O_j(0)\rangle = \frac{C_{ij}\,\delta_{\Delta_i,\Delta_j}}{|x|^{2\Delta_i}}.$$

With an orthonormal basis of scalar primaries, one often sets $C_{ij}=\delta_{ij}$. In holography, this normalization choice changes the normalization of the corresponding bulk fields and cubic couplings.

For vectors, define

$$I_{ij}(x)=\delta_{ij}-2\frac{x_i x_j}{x^2}.$$

A vector primary has

$$\langle V_i(x)V_j(0)\rangle = \frac{C_V I_{ij}(x)}{|x|^{2\Delta}}.$$

For a conserved current, $\Delta=d-1$:

$$\langle J_i(x)J_j(0)\rangle = \frac{C_J I_{ij}(x)}{|x|^{2(d-1)}}.$$

For the stress tensor,

$$\langle T_{ij}(x)T_{kl}(0)\rangle = \frac{C_T}{|x|^{2d}}\mathcal I_{ij,kl}(x),$$

where

$$\mathcal I_{ij,kl}(x) = \frac12\left(I_{ik}I_{jl}+I_{il}I_{jk}\right) - \frac{1}{d}\delta_{ij}\delta_{kl}.$$

The coefficient $C_T$ measures the number of local degrees of freedom. In a holographic CFT with an Einstein-gravity dual,

$$C_T\sim \frac{L^{d-1}}{G_N}.$$

For large-$N$ adjoint gauge theories, typically $C_T\sim N^2$.

Three-point functions

For scalar primaries, conformal symmetry fixes the position dependence:

$$\langle \mathcal O_1(x_1)\mathcal O_2(x_2)\mathcal O_3(x_3)\rangle = \frac{C_{123}} {|x_{12}|^{\Delta_1+\Delta_2-\Delta_3} |x_{13}|^{\Delta_1+\Delta_3-\Delta_2} |x_{23}|^{\Delta_2+\Delta_3-\Delta_1}}.$$

The coefficient $C_{123}$ is dynamical CFT data. In a classical holographic theory, it is computed by a cubic bulk coupling:

$$C_{123} \longleftrightarrow \int_{\mathrm{AdS}} K_{\Delta_1}K_{\Delta_2}K_{\Delta_3}\times \lambda_{123}.$$

For spinning operators, three-point functions have a finite number of allowed tensor structures. Conservation, parity, supersymmetry, and global symmetries may reduce the number of independent coefficients.

Four-point functions and cross ratios

For four scalar primaries, conformal symmetry does not fix the whole answer. It reduces the answer to a function of cross ratios. For identical scalars of dimension $\Delta$,

$$\langle \mathcal O(x_1)\mathcal O(x_2)\mathcal O(x_3)\mathcal O(x_4)\rangle = \frac{1}{|x_{12}|^{2\Delta}|x_{34}|^{2\Delta}}\mathcal G(u,v),$$

where

$$u= \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \qquad v= \frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}.$$

Here the first variable is the letter $u$. It is visually close to the Greek letter $\nu$, so in typed notes it is worth checking this carefully.

Crossing symmetry for identical scalars gives

$$\boxed{ v^\Delta \mathcal G(u,v) = u^\Delta \mathcal G(v,u). }$$

This is the simplest bootstrap equation. In a holographic CFT, the large-$N$ expansion of $\mathcal G(u,v)$ is the boundary avatar of tree-level and loop-level bulk Witten diagrams.

The operator product expansion

The OPE says that nearby operators can be replaced by a sum of local operators:

$$\mathcal O_i(x)\mathcal O_j(0) \sim \sum_k C_{ij}{}^k(x,\partial)\mathcal O_k(0).$$

For scalar primaries, the leading primary contribution is schematically

$$\mathcal O_i(x)\mathcal O_j(0) \sim \sum_k \frac{C_{ijk}}{|x|^{\Delta_i+\Delta_j-\Delta_k}} \left[\mathcal O_k(0)+\text{descendants}\right].$$

The descendants are fixed by conformal symmetry. The numbers $C_{ijk}$ are the same OPE coefficients that appear in three-point functions, after choosing two-point normalizations.

The OPE is not just a short-distance heuristic. In Euclidean CFT, inside correlation functions, it has a finite radius of convergence set by the nearest other operator insertion.

Conformal block expansion

In a four-point function, the OPE decomposes $\mathcal G(u,v)$ into conformal blocks:

$$\mathcal G(u,v) = \sum_{\mathcal O} \lambda_{\phi\phi\mathcal O}^2 G_{\Delta,\ell}(u,v),$$

where the sum is over primary operators appearing in the $\phi\times\phi$ OPE. The conformal block $G_{\Delta,\ell}$ contains the contribution of a primary and all descendants in its conformal multiplet.

The bootstrap equation is schematically

$$\sum_{\mathcal O} \lambda_{\phi\phi\mathcal O}^2 \left[ v^{\Delta_\phi}G_{\Delta,\ell}(u,v) - u^{\Delta_\phi}G_{\Delta,\ell}(v,u) \right]=0.$$

For holography, conformal block decompositions are useful because exchange Witten diagrams decompose into single-trace and double-trace conformal blocks. Bulk locality is encoded in the detailed pattern of these CFT contributions.

Large $N$ CFTs and generalized free fields

In a large-$N$ holographic CFT, normalized single-trace operators often satisfy

$$\langle \mathcal O\mathcal O\rangle\sim N^0, \qquad \langle \mathcal O\mathcal O\mathcal O\rangle_c\sim \frac1N, \qquad \langle \mathcal O^n\rangle_c\sim N^{2-n}.$$

At leading order, many single-trace operators behave as generalized free fields. Their four-point functions factorize into products of two-point functions:

$$\langle \mathcal O_1\mathcal O_2\mathcal O_3\mathcal O_4\rangle = \langle \mathcal O_1\mathcal O_2\rangle \langle \mathcal O_3\mathcal O_4\rangle + \text{two pairings} + O(1/N^2).$$

The corresponding double-trace operators have approximate dimensions

$$\Delta_{n,\ell} = \Delta_1+\Delta_2+2n+\ell+\gamma_{n,\ell},$$

with

$$\gamma_{n,\ell}=O(1/N^2)$$

for weakly interacting bulk fields. This is the CFT origin of bulk Fock space.

Relevant, marginal, and irrelevant deformations

A deformation

$$S\to S+\int d^d x\,g\mathcal O$$

has coupling dimension

$$[g]=d-\Delta.$$

Thus:

operator dimension	name	RG behavior near the fixed point
$\Delta<d$	relevant	grows toward the IR
$\Delta=d$	marginal	needs beta-function analysis
$\Delta>d$	irrelevant	suppressed in the IR

In AdS/CFT, a relevant deformation changes the boundary condition for a bulk field and often drives the geometry away from AdS in the interior. Domain-wall geometries are the gravitational representation of RG flows.

A marginal operator can be exactly marginal, marginally relevant, or marginally irrelevant. In $\mathcal N=4$ SYM, the complexified gauge coupling is exactly marginal.

Conserved currents

A conserved global current obeys

$$\partial_\mu J^\mu=0.$$

In a unitary CFT, a conserved spin-one current has protected dimension

$$\Delta_J=d-1.$$

It couples to a background gauge field:

$$\delta S=\int d^d x\sqrt g\,A_\mu J^\mu.$$

The source dimension is

$$[A_\mu]=1.$$

In AdS/CFT,

$$J^\mu \quad\longleftrightarrow\quad A_M^{\rm bulk}.$$

The boundary current is global from the CFT perspective. The bulk gauge field is dynamical in the gravitational theory, but its boundary value is a nondynamical source unless one explicitly gauges the boundary symmetry.

Stress tensor

The stress tensor obeys

$$\partial_\mu T^{\mu\nu}=0, \qquad T^\mu{}_{\mu}=0$$

in flat space for an undeformed CFT, up to anomalies and contact terms. It has protected dimension

$$\Delta_T=d.$$

It couples to the background metric:

$$\delta S =\frac12\int d^d x\sqrt g\,T^{\mu\nu}\delta g_{\mu\nu}.$$

The holographic map is

$$T_{\mu\nu} \quad\longleftrightarrow\quad \text{bulk metric }g_{MN}.$$

The coefficient $C_T$ in the two-point function is proportional to the inverse bulk Newton constant:

$$C_T\propto\frac{L^{d-1}}{G_N}.$$

In even-dimensional CFTs, the stress-tensor trace on a curved background can have a Weyl anomaly:

$$\langle T^\mu{}_{\mu}\rangle=\mathcal A[g].$$

For a two-dimensional CFT, one common convention is

$$\langle T^i{}_i\rangle=-\frac{c}{24\pi}R,$$

with sign depending on curvature and stress-tensor conventions.

Ward identities

The Euclidean generating functional is written here as

$$Z[g,J,A]=e^{-W[g,J,A]}.$$

One-point functions are obtained by variation:

$$\langle\mathcal O\rangle =\frac{1}{\sqrt g}\frac{\delta W}{\delta J},$$ $$\langle J^\mu\rangle =\frac{1}{\sqrt g}\frac{\delta W}{\delta A_\mu},$$ $$\langle T^{\mu\nu}\rangle =\frac{2}{\sqrt g}\frac{\delta W}{\delta g_{\mu\nu}}.$$

Different sign conventions for $W$ and source insertions change signs. The reliable rule is to check the source term in the action.

For a scalar source $J$ coupled to $\mathcal O$, diffeomorphism invariance gives schematically

$$\nabla_\mu\langle T^\mu{}_{\nu}\rangle = \langle\mathcal O\rangle\nabla_\nu J +F_{\nu\mu}\langle J^\mu\rangle+ \cdots.$$

Weyl invariance gives

$$\langle T^\mu{}_{\mu}\rangle =(\Delta-d)J\langle\mathcal O\rangle +\mathcal A[g,J,A]$$

when sources are present. For an undeformed flat-space CFT, the right-hand side vanishes.

These are the boundary versions of radial constraint equations in holographic renormalization.

Two-dimensional CFT special case

In two Euclidean dimensions, use complex coordinates

$$z=x^1+ix^2, \qquad \bar z=x^1-ix^2.$$

A primary has holomorphic and antiholomorphic weights $(h,\bar h)$, with

$$\Delta=h+\bar h, \qquad s=h-\bar h.$$

The two-point function is

$$\langle\mathcal O(z,\bar z)\mathcal O(0,0)\rangle =\frac{C_{\mathcal O}}{z^{2h}\bar z^{2\bar h}}.$$

The local conformal algebra becomes two copies of Virasoro:

$$[L_m,L_n] =(m-n)L_{m+n}+\frac{c}{12}m(m^2-1)\delta_{m+n,0},$$

and similarly for $\bar L_n$ with central charge $\bar c$.

For AdS$_3$/CFT$_2$ with parity invariance,

$$c=\bar c=\frac{3L}{2G_3}.$$

This infinite-dimensional structure is special to two boundary dimensions. Do not assume Virasoro methods automatically generalize to higher-dimensional CFTs.

Euclidean versus Lorentzian CFT

Most conformal kinematics is cleanest in Euclidean signature. Radial quantization and reflection positivity then reconstruct a unitary Lorentzian theory under suitable assumptions.

The Lorentzian theory contains causal information absent from Euclidean kinematics alone. For example:

• Wightman functions depend on operator ordering;
• retarded functions encode response;
• commutators vanish at spacelike separation;
• singularities have $i\epsilon$ prescriptions;
• thermal correlators satisfy KMS relations.

In holography, Euclidean regularity gives Euclidean correlators, while Lorentzian retarded correlators require infalling boundary conditions at horizons.

Embedding-space shorthand

For many conformal-kinematic formulas, one embeds $d$-dimensional points into a projective null cone in $\mathbb R^{d+1,1}$. A boundary point $x^i$ is represented by a null vector $P^A(x)$ satisfying

$$P^2=0, \qquad P\sim\lambda P.$$

Scalar correlators become homogeneous functions of dot products $P_i\cdot P_j$. The advantage is that conformal transformations act linearly as $SO(d+1,1)$ transformations in Euclidean signature.

This course rarely needs embedding space explicitly, but many CFT and Witten-diagram references use it.

CFT data to bulk dictionary

CFT quantity	Bulk meaning
scalar primary $\mathcal O$	scalar field $\phi$
dimension $\Delta$	mass through $m^2L^2=\Delta(\Delta-d)$
spin $\ell$ primary	spin-$\ell$ bulk field
conserved current $J^\mu$	bulk gauge field $A_M$
stress tensor $T_{\mu\nu}$	bulk metric fluctuation $h_{MN}$
$C_T$	$L^{d-1}/G_N$ up to convention
single-trace operator	single-particle bulk state
double-trace operator	two-particle bulk state
OPE coefficient $C_{ijk}$	cubic bulk coupling after normalization
anomalous double-trace dimensions	bulk binding/scattering data
large $N$ factorization	weak bulk interactions
large single-trace gap	local bulk EFT below the string scale

This table is a guide, not a theorem in isolation. The full statement requires a consistent large-$N$ CFT with the right spectrum and OPE data.

Normalization traps

Two-point normalization changes OPE coefficients

If

$$\mathcal O_i\to a_i\mathcal O_i,$$

then

$$C_{ijk}\to a_i a_j a_k C_{ijk}.$$

Only after fixing two-point normalizations do OPE coefficients become directly comparable.

$C_T$ conventions vary

The coefficient $C_T$ is defined with different numerical normalizations in different communities. Always compare the complete two-point-function convention, not just the symbol $C_T$.

Contact terms are not optional bookkeeping

Ward identities often contain contact terms. In momentum space, contact terms appear as polynomials in momenta. They can shift local pieces of correlators and are tied to counterterm choices in holography.

Conserved currents are global currents in the boundary theory

A bulk gauge field is dual to a global symmetry current in the boundary CFT. To make the boundary symmetry dynamical, one must gauge it by adding boundary degrees of freedom or integrating over the boundary gauge field.

“Primary” does not mean “elementary”

A primary operator is defined representation-theoretically by its transformation under conformal symmetry. It can be elementary in a Lagrangian theory, composite, non-Lagrangian, or an abstract operator in a bootstrap description.

Mini-reference: formulas to remember

Scalar two-point function:

$$\langle\mathcal O(x)\mathcal O(0)\rangle =\frac{C_{\mathcal O}}{|x|^{2\Delta}}.$$

Scalar three-point function:

$$\langle\mathcal O_1\mathcal O_2\mathcal O_3\rangle = \frac{C_{123}} {|x_{12}|^{\Delta_1+\Delta_2-\Delta_3} |x_{13}|^{\Delta_1+\Delta_3-\Delta_2} |x_{23}|^{\Delta_2+\Delta_3-\Delta_1}}.$$

Cross ratios:

$$u= \frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2}, \qquad v= \frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}.$$

OPE:

$$\mathcal O_i(x)\mathcal O_j(0) \sim \sum_k \frac{C_{ijk}}{|x|^{\Delta_i+\Delta_j-\Delta_k}} \left[\mathcal O_k(0)+\text{descendants}\right].$$

State-operator map:

$$|\mathcal O\rangle=\mathcal O(0)|0\rangle, \qquad H_{\rm cyl}|\mathcal O\rangle=\Delta|\mathcal O\rangle.$$

Bulk mass-dimension relation:

$$m^2L^2=\Delta(\Delta-d).$$

Unitarity bounds:

$$\Delta_{\rm scalar}\ge\frac{d-2}{2}, \qquad \Delta_{\ell\ge1}\ge\ell+d-2.$$

Dictionary checkpoint

The CFT side of AdS/CFT is not just “some boundary theory.” The precise boundary input is CFT data,

$$\{\Delta_i,\ell_i,C_{ijk}\},$$

plus global-symmetry, anomaly, contact-term, and large-$N$ structure. The simplest holographic translation is

$$\Delta_i \leftrightarrow \text{bulk masses}, \qquad C_{ijk}\leftrightarrow \text{bulk interaction vertices}, \qquad \frac1N\leftrightarrow \text{bulk coupling}.$$

A large-$N$ CFT with a sparse low-dimension single-trace spectrum is the natural boundary candidate for a local semiclassical bulk dual.

Common confusions

“Conformal symmetry fixes all correlators.”

It fixes two- and three-point position dependence, but not the operator spectrum or OPE coefficients. Four-point functions contain nontrivial functions of cross ratios. The dynamical content of a CFT is precisely the allowed CFT data.

“Scale invariance and conformal invariance are always the same.”

In many familiar unitary relativistic QFTs, scale invariance plus additional assumptions implies conformal invariance, but this is not a purely automatic identity in all possible theories and dimensions. AdS/CFT uses conformal invariance because the AdS isometry group matches the conformal group.

“A primary operator is a fundamental field.”

No. A primary is the top state of a conformal multiplet. It can be elementary in a Lagrangian theory, composite, non-Lagrangian, or an abstract operator in a bootstrap description.

“A conserved current is just another spin-one operator.”

Conservation shortens the conformal multiplet and fixes the dimension to $\Delta=d-1$. In holography, this shortening is reflected by bulk gauge redundancy.

“A marginal operator is automatically exactly marginal.”

No. A dimension-$d$ operator at a fixed point is marginal at first order. It may become marginally relevant, marginally irrelevant, or exactly marginal depending on beta functions and symmetry constraints.

“CFT central charge always means the same thing.”

In two dimensions, $c$ is the Virasoro central charge. In higher dimensions, people use coefficients such as $C_T$, $a$, and $c$ depending on context. Holographic formulas depend on the normalization.

Exercises

Exercise 1: Derive the scalar two-point scaling

Use scale invariance to show that a scalar-primary two-point function must scale as $|x|^{-2\Delta}$ if the two operators have the same dimension.

Solution

Let

$$G(x)=\langle\mathcal O(x)\mathcal O(0)\rangle.$$

Under $x\to\lambda x$, a scalar primary contributes one factor of $\lambda^{-\Delta}$ at each insertion. Thus

$$G(\lambda x)=\lambda^{-2\Delta}G(x).$$

Rotational invariance implies $G(x)=G(|x|)$, so

$$G(x)=\frac{C_{\mathcal O}}{|x|^{2\Delta}}.$$

Special conformal transformations further imply that scalar primaries of different dimensions have vanishing two-point functions in a basis of scaling operators.

Exercise 2: Source dimension

A scalar operator has dimension $\Delta$. What is the dimension of its source $J$ in

$$\int d^d x\,J\mathcal O?$$

Solution

The action is dimensionless in units where $\hbar=1$. Since

$$[d^d x]=-d, \qquad [\mathcal O]=\Delta,$$

we require

$$[J]+\Delta-d=0.$$

Thus

$$[J]=d-\Delta.$$

This is why relevant operators with $\Delta<d$ have positive-dimension couplings.

Exercise 3: Current dimension from conservation

Use the spin-$\ell$ unitarity bound

$$\Delta\ge \ell+d-2$$

for $\ell\ge1$ to determine the dimension of a conserved current and a conserved stress tensor.

Solution

For a spin-one current, $\ell=1$, saturation gives

$$\Delta_J=1+d-2=d-1.$$

The shortening condition is current conservation:

$$\partial_\mu J^\mu=0.$$

For the stress tensor, $\ell=2$, saturation gives

$$\Delta_T=2+d-2=d.$$

The corresponding shortening condition is stress-tensor conservation:

$$\partial_\mu T^{\mu\nu}=0.$$

Exercise 4: Double-trace dimensions

Two scalar single-trace operators have dimensions $\Delta_1$ and $\Delta_2$. In a generalized-free large-$N$ limit, what are the approximate dimensions of the double-trace primaries with spin $\ell$ and radial excitation number $n$?

Solution

The approximate double-trace primaries are schematically

$$[\mathcal O_1\mathcal O_2]_{n,\ell} \sim \mathcal O_1\partial^{2n}\partial_{\{\mu_1}\cdots\partial_{\mu_\ell\}}\mathcal O_2+\cdots,$$

where the dots enforce primary and traceless conditions. Their dimensions are

$$\Delta_{n,\ell} = \Delta_1+\Delta_2+2n+\ell+\gamma_{n,\ell}.$$

At leading generalized-free order,

$$\gamma_{n,\ell}=0.$$

In a weakly interacting holographic CFT,

$$\gamma_{n,\ell}=O(1/N^2),$$

encoding bulk binding and scattering effects.

Further reading

• Slava Rychkov, EPFL Lectures on Conformal Field Theory in $D\ge3$ Dimensions.
• David Simmons-Duffin, TASI Lectures on the Conformal Bootstrap.
• P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory.
• H. Osborn and A. Petkou, Implications of Conformal Invariance in Field Theories for General Dimensions.
• S. Ferrara, R. Gatto, and A. F. Grillo, classic work on conformal representations and unitarity bounds.
• G. Mack, classic work on conformal representations and unitarity bounds.